Maths Statistics flashcards

Types of random sampling

• simple random sampling

• systematic sampling

• stratified sampling

Types of non-random sampling

• quota sampling

• opportunity sampling

Simple random sampling

• Every sample has an equal chance ofbeing selected

• Each item has an idenitfying number

• Random number generator can be used

Advantages of simple random sampling

• Bias free

• Easy and cheap to implement

• Each number has a known equal chance of being selected

Disadvantages of simple random sampling

• Not suitable when population size is large

• Sampling frame needed

Systematic sampling

• Required elements are chosen at regular intervals in ordered list

• I.e. take every kth elements where k=population size/sampling size starting at random item between 1 and k

Advantages of systematic sampling

• Simple and quick to use

• Suitable for large samples/population

Disadvantages of systematic sampling

• Sampling frame again needed

• Can introduce bias if sampling frame not random

Stratified sampling

• Population divided into groups (strata) and a simple random sample carried out in each group

• Same proportion sampling size/population size sampled from each strata

• Used when sample is large and population naturally divides into groups.

Advantages of stratified sampling

• Sample accurately reflects the population structure

• Guarantees proportional representation of groups within a population

Disadvantages of stratified sampling

• Population must be clearly classified into distinct strata

• Selection within each stratum suffers from the same disadvantages as simple random sampling

Quota sampling

• Interviewer or researcher selects a sample that reflects the characteristics of the whole popoulation

• A quota of items/people in each group is set to try and reflect the group’s proportion in the whole population 

• Interviewer selects the actual sampling units

Advantages of quota sampling

• Allows a small sample to still be representative of the population

• No sampling frame frquired

• Quick, easy and inexpensive

• Allows for easy comparison between different groups within a population

Disadvantages of quota sampling

• Non-random sampling can introduce bias

• Population must be divided into groups, which can be costly or inaccurate

• Increasing scope of study increases number of groups, which adds time and expense

• Non-responses are not recorded as such

Opportunity sampling

Sample taken from people who are available at time of study, who meet the criteria

Advantages of opportunity sampling

• Easy to carry out

• inexpensive

Disadvantages of opportunity sampling

• Unlikely to provide a representative sample

• Highly dependant on individual researcher

Qualitative/categorical data

non-numerical value data, e.g. colour

Quantitative data

numerical value data

Discrete data

can only take specific values - e.g. shoe size

continuous data

can take any decimal value within a specific range

x

variable that represents the value of multiple objects (i.e. a bit like a set) - in stats

∑x

sum of the data set

𝑥̄ (x-bar)

mean of the data set

Median

the middle value when the data values are put in order

• (n+1) / 2 = median no.

  • If odd - use that number for median

  • If even - use integer above and below for median

Formula for mean

𝑥̄ = ∑x / n

• n - the number of data values

• FX-XG50 - statistics, put in values in list 1, calc, 1-var

or

𝑥̄ = ∑ƒx / ∑ƒ

• f - the frequency

• n - number of data values

• For ungrouped formula

Linear interpolation

process for estimating the median

Median = lower class boundary + ((n/2)-C.F)) x class width

• C.f. being class frequency

Median (second quartile)

at the 50% point (written as Q2)

Lower quartile

Lower quartile - 25% into data set (written as Q1)

Upper quartile

Upper quartile - 75% into data set (written as Q2)

Finding lower quartile in discrete data set

Multiply n by ¼ 

• If whole LQ is between this value and the one above

• If not whole round up and take that data point

Finding upper quartile in discrete data set

Multiply n by ¾ 

• If whole UQ is between this value and the one above

• If not whole round up and take that data point

Finding lower quartile in continuous data set

Divide n by 4 and take that data point

Finding upper quartile in continuous data set

Divide 3n by 4 and take that data point

Formula for percentiles or quartiles

LB + ((PL/GF) x CW)

• LB - lower quartile boundary

• PL - (LB - CM) of previous - places into the group

• GF - group frequency 

• CW - class width

Variance

Average squared distance from the mean, measure of spread that takes into account all values

Variance formula

σ² = x - x̄

Or

σ² = (Σx²/n) - x²

• σ² = variance

Standard deviation

the value’s distance from the mean

standard deviation formula

σ = √variance

• σ = standard deviation

Standard deviation rules

∑(x₁ - x̄) = 0 - the sum of standard deviations from the mean is 0

Outlier

an extreme value, usually 1.5 IQRs beyond the lower and upper quartiles

Graphing cumulative frequency

take upper boundary (x-axis) and frequency (y-axis) for the points and join them with a curve

Histogram information

• Area of each bar is proportional to the frequency for each group

• Histograms for continuous data

• Frequency polygon - midpoints of the histogram density graph joined up

Frequency density formula

Frequency density = frequency / classwidth

Histogram Area Formula

Area = k x Frequency